Diffraction and PolarizationDiffractionDiffraction from narrow slitsResolution issuesThe diffraction gratingPolarization

Diffraction

Light rays can bend around corners

No sharp boundary exists between the dark and bright parts on the screen

Fraunhofer

(far field) Diffraction

The rays from the slit are parallel to each other.

Either the screen is far from the slit or a lens is used to focus the light rays from the slit.

Diffraction From Narrow SlitsEach portion of the slit acts as a wave source.

2sin

2λθ =

a

aλθ =sin

aλθ 2sin =

aλθ 3sin =

Divide slit into two parts

Now divide it into four parts

And into six parts

… so in general for destructive interferencea

mλθ =sin ,...3,2,1 ±±±=m

am

mλθ =sin

Lym

m =θtan

θθ sintan ≈

Ly

am m≈λ

maLym λ≈

,...3,2,1 ±±±=m

Positions of Dark Fringes

Intensity of the Single Slit Diffraction Pattern

Each Δy contributes an electric field of ΔE for the total field at P.

When adding the fields, we should take into account the relative phase differences of the fields.

θλπβ sin2 yΔ=Δ

Adding up all these fields results in:

More accurately, the pattern is a result of the interference of the waves coming from different parts of the slit.

( ) 2

max sinsinsin

⎥⎦

⎤⎢⎣

⎡=

λθπλθπ

aaII

Two-Slit Diffraction PatternInterference and diffraction occur simultaneously

( ) 22

max sinsinsinsincos ⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛=

λθπλθπ

λθπ

aadII

Resolution of Single Slit and Circular Apertures

When the central maximum of one image falls on the first minimum of the other Image, the images are said to be just resolved. This limiting condition is called the Rayleigh’s Criterion.

Rayleigh’s Criterion

aλθ =sin

aλθ =min

Dλθ 22.1min =For circular apertures

For a given lens diameter, which light gives the best resolution in a microscope?

1. red2. yellow3. green4. blue5. All give the same resolution.

Concept Question

Limiting Resolution of a Microscope

cmDnm

9.0589

==λ rad

D5

2

9

min 1098.7109.01058922.122.1 −

−

−

×=××

==λθ

nm400=λ rad52

9

min 1042.5109.01040022.1 −

−

−

×=××

=θ

33.1=watern

rad

nmnmnwater

vacuumwater

52

9

min 106109.01044322.1

44333.1

589

−−

−

×=××

=

===

θ

λλ

SpectroscopyFor many experiments we would like to analyze the variation of light intensity as a function of wavelength.This light can be from a fluorescent source or light transmitted through a sample.A spectrometer will disperse light according to its wavelength and detect each wavelength component separately.

Diffraction GratingsA device with closely spaced slits (or grooves).Light passing through the slits will interfere with each other to produce a multiple slit interference pattern The position of the constructive peaks will be dependent on wavelength.

λθ md =sin

Spectrometers

1. red spots at 0° and ±45°.2. yellow spots at 0° and ±45°.3. orange spots at 0° and ±45°.4. an orange spot at 0°, yellow spots at ±45°, and red spots slightly

farther out.5. an orange spot at 0°, yellow spots at ±45°, and red spots slightly

closer in.

Concept QuestionA diffraction grating is illuminated with yellow light at normal incidence. The pattern seen on a screen behind the grating consists of three yellow spots, one at zero degrees (straight through) and one each at ±45°.You now add red light of equal intensity, coming in the same direction as the yellow light. The new pattern consists of

Problem 38.24

2 3

Nd 1=

dmλθ =sin

θ( ) ( )nmN

dnm

v 40034003sin 3 ==θ

nmnm

r

v

600400

==

λλ

( ) ( )nmNd

nmr 60026002sin 2 ==θ

23

23 sinsin

rv

rv

θθθθ

==

N

X-Ray DiffractionAn X-ray has a wavelength on the order of the atomic separation in crystals (~0.1 nm).Successive layers of atoms act as a diffraction grating for the X-ray and create an interference pattern.The pattern is a characteristic of the crystal structure.

,...3,2,1sin2=

=m

md λθ

Polarization of Light Waves

An EM wave linearly polarized in the y direction

An unpolarized fieldA linearly polarized field

θ2max cosII =

Malus’ Law

Polarization by Selective Absorption

These polarizers work by transmitting light along their transmission axis.The transmitted light will have a polarization along the transmission axis.Unpolarized light will be transmitted as a linearly polarized wave at half the intensity.

When a ray of light is incident on two polarizers with their polarization axes perpendicular, no light is transmitted. If a third polarizer is inserted between these two with its polarization axis at 45° to that of the other two, does any light get through to point P?

1. yes2. no

Concept Question

Polarization by ReflectionThe polarization of reflected and refracted rays are dependent on the angle of incidence. For one angle, the reflected light is completely linearly polarized.

( ) pp

p

θθθ

θθ

cos90sinsin

90

2

2

=−=

°=+

p

ppnθθ

θθ

θθ

cossin

sinsin

sinsin

22

1 ===

pn θtan=Brewster’s Angle

Polarization by Double RefractionIn an amorphous material: n is the same for all polarizationsIn some crystalline materials: n depends on the direction of polarization

Birefringent materials: ordinary (no ) and extra-ordinary (ne ) indices of refraction

Polarization by Scattering

Examples:Blue skies, red sunsets

Optical ActivityOptically active materials are materials that rotate the polarization of any light transmitted through them.This is due to molecular asymmetry which can be caused by the molecular structure or external factors such as mechanical stress.

For Next ClassReview for the Final on FridayWebAssign: Assignment 16Final on Monday, 9:00 am